ORIGINAL ARTICLE

Predictive modeling the side weir discharge coefficient usingneural network

Abbas Parsaie1

Received: 16 January 2016 / Accepted: 18 March 2016 / Published online: 2 April 2016

� Springer International Publishing Switzerland 2016

Abstract Weirs are common structures that are widely

used in the almost water engineering projects such as

hydropower systems, irrigation and drainage networks

along with sewage networks. A side weir has many pos-

sible uses in hydraulic engineering and has been investi-

gated as an important structure in hydro systems, as well.

In this paper, predicting the discharge coefficient of side

weirs (Cdsw) was considered using the empirical formulas,

multilayer perceptron (MLP) and radial basis function

(RBF) neural network as delegate of artificial neural net-

work models. The results indicate that Emiroglu formula

by correlation coefficient of (R2 = 0.65) and root mean

square error of (RMSE = 0.03) is the accurate one among

the empirical formulas. Evaluating the performance of the

RBF model with ten neurons in the hidden layer involving

error indices of (R2 = 0.71 and RMSE = 0.08) showed

that this model was a bit better than Emiroglu formula. The

structure of MLP model was considered as similar to RBF

model whereas the tangent sigmoid was used instead to the

radial basis function. The results of MLP model showed

that this model with R2 = 0.89 and RMSE = 0.067 had

suitable performance for predicting discharge coefficient.

Performance of MLP was more accurate compared to RBF

model.

Keywords Weirs � Discharge coefficient � Empirical

formulas � Multi-layer perceptron (MLP)

Introduction

Modeling of hydraulic structure has received much atten-

tion in recent years due to its effect on increasing hydro

system performance (Dehdar-behbahani and Parsaie 2016;

Parsaie et al. 2015b). Weirs are common structures, which

are widely used in most water engineering projects such as

hydropower systems, irrigation and drainage networks and

sewage networks (Haghiabi 2012). Side weir is a type of

weir which has many possible uses in hydraulic engineering

and has been investigated as an important structure in hydro

systems as well (Chen 2015; Laycock 2007). Side weir is a

hydraulic structure placed on the side of the channel and

sometime is used as water surface controller structure in

dam and irrigation projects whereas the main task of side

weirs is removing the excess flow from the hydro systems

(Bagheri et al. 2014; Haddadi and Rahimpour 2012). Study

on side weirs hydraulics is conducted by the physical and

numerical approaches (Parsaie 2016). In the field of phys-

ical studies, researchers attempted to improve the perfor-

mance of side weirs by proposing various shapes for the

crest of the side weir and compared its performance with

rectangular shape as standard form for side weir. In this

regard, labyrinth, oblique, semi-elliptical, curved plan-form

and trapezoidal sharp and broad-crested could be men-

tioned. Based on reports, performance of nonlinear weirs is

much more than the conventional side weir (Borghei and

Parvaneh 2011; Cheong 1991; Cosar and Agaccioglu 2004;

Emiroglu and Kaya 2011; Emiroglu and Kisi 2013; Emir-

oglu et al. 2011a; Haddadi and Rahimpour 2012; Jalili and

Borghei 1996; Kaya et al. 2011). In the field of numerical

modeling, in addition to solving the governing hydraulic

equations by numerical approaches such as Runge–Kutta

Method, the computational fluid dynamic (CFD) techniques

has been used to simulate the flow over side weir.

& Abbas Parsaie

abbas_parsaie@yahoo.com

1 Department of Water Engineering, Lorestan University,

Khorramabad, Iran

123

Model. Earth Syst. Environ. (2016) 2:63

DOI 10.1007/s40808-016-0123-9

Numerical solution of governing equations leads to defining

hydraulic parameters such as water surface profile, distri-

bution of velocity and pressure and flow pattern (Aydin and

Emiroglu 2013; Parsaie and Haghiabi 2015b). Another way

for numerical modeling is related to the use of soft com-

puting techniques for predicting the hydraulic properties of

side weirs such as discharge coefficient (Azamathulla et al.

2016). In this regard, researchers used artificial neural net-

work (ANNs), group method of data handling (GMDH),

gene expression programming (GEP), and adaptive neuro-

fuzzy inference system (ANFIS). Developing ANN models

is based on data set, which means that to predict the

hydraulic phenomenon by neural network techniques,

parameters that influence the phenomenon should be mea-

sured in the past. ANN models could be used as standalone

and be applied as participant of numerical methods in

numerical simulation as well to increase the accuracy of the

numerical modeling. The results of using the mentioned

neural network models indicate that ANN models are more

accurate (Bilhan et al. 2010; Bilhan et al. 2011; Ebtehaj

et al. 2015a, b; Emiroglu and Kisi 2013; Emiroglu et al.

2011b; Kisi et al. 2012; Parsaie and Haghiabi 2015b). In this

research, the radial basis function (RBF) neural network

that has high performance in pattern recognition and image

processing is used for predicting the side weir discharge

coefficient, and its performance is compared with empirical

formula and multilayer perceptron neural network as com-

mon ANN model, which is used by most researchers.

Materials and method

Figure 1 shows a schematic shape of a side weir in sub-

critical flow condition. The discharge coefficient of side

weir is a function of hydraulic characteristics and geometry

of side weir and main channel. Most hydraulic and

geometry parameters are shown in Fig. 1.

As could be seen in Fig. 1, V1 and Q1 are the velocity

and discharge of flow at beginning the side weir

respectively; B: width of main channel; E: specific

energy; h1: depth of flow at beginning the side weir; h:

the flow over side weir; h2: depth of flow at the end of

side weir; L: side weir length; P: weir height and the

longitudinal slope of the channel (S0). Defining the

effect of each effective parameter requires conducting

experiments in the condition that other parameters are

constant. Researchers who conducted experimental

studies on the hydraulics of side weirs proposed empir-

ical equations for calculating the side weir discharge

coefficient. A summary of the most famous empirical

formulas is given in Table 1.

Researchers attempt to reduce the number of exper-

iments using the dimensional analysis techniques such

as Buckingham p theory. Using analysis techniques

leads to derive dimensionless parameters. Dimension-

less parameters which influence the discharge coeffi-

cient of side weirs are given in Eq. (1) (Emiroglu et al.

2011a).

Cdsw ¼ f2 Fr1;L

B;L

h1;P

h1

� �ð1Þ

In Eq. (1), Fr1 is the Froude number, LBdescribes the

ratio of weir length to the width of main channel, Lh1

describes the ratio of weir length to the flow depth at

beginning the weir, and Ph1

describes the ratio of weir

height to the flow depth at the beginning the weir. As

presented in Table 1, most empirical formulas have used

dimensionless parameters. Using dimensionless parame-

ters in ANN model preparation leads to developing

optimal structure. Developing MLP and RBF models

Fig. 1 Sketch of side weir

structure in subcritical flow

condition

63 Page 2 of 11 Model. Earth Syst. Environ. (2016) 2:63

123

similar to other neural network models is based on data

set. To do so, 477 data sets related to side weir discharge

coefficient, published in creditable journals were col-

lected. Some of resources used for data derivation are

given as follows (Bagheri et al. 2014; Borghei et al. 1999;

Emiroglu et al. 2011a; Singh et al. 1994; Subramanya and

Awasthy 1972). The range of collected data is given

Table 2.

Multilayer perceptron (MLP) neural network

ANN is a nonlinear mathematical model that is able to

simulate arbitrarily complex nonlinear processes, which

relate inputs and outputs of any system. In many complex

mathematical problems that lead to solving complex non-

linear equations, Multilayer perceptron networks are com-

mon types of ANN widely used by researchers. To use MLP

model, definition of appropriate functions, weights and bias

should be considered. Due to the nature of the problem,

different activity functions in neurons could be used. An

ANN may have one or more hidden layers. Figure 2

demonstrates a three-layer neural network consisting of

inputs layer, hidden layer (layers) and outputs layer. As

shown in Fig. 2, nwi is the weight and bi is the bias for each

neuron. Weight and biases values will be assigned progres-

sively and corrected during training process comparing the

predicted outputs with known outputs. Such networks are

Table 1 Some empirical formulas to calculate side weir discharge coefficient

Row Author Equation

1 Nandesamoorthy and Thomson

(1972)Cd ¼ 0:432

2�Fr21

1þ 2Fr21

� �0:5

2 Subramanya and Awasthy (1972)Cd ¼ 0:864

1�Fr21

2þFr21

� �0:5

3 Yu-Tech (1972) Cd ¼ 0:623 � 0:222Fr1

4 Ranga Raju et al. (1979) Cd ¼ 0:81 � 0:6Fr1

5 Hager (1987)Cd ¼ 0:485

2�Fr21

2þ 3Fr21

� �0:5

6 Cheong (1991) Cd ¼ 0:45 � 0:221Fr1

7 Singh et al. (1994) Cd ¼ 0:33 � 0:18Fr1 þ 0:49 Ph1

� �

8 Jalili and Borghei (1996) Cd ¼ 0:71 � 0:41Fr1 þ 0:22 Ph1

� �

9 Borghei et al. (1999) Cd ¼ 0:7 � 0:48Fr1 þ 0:3 Ph1

� �þ 0:06 L

h1

� �

10 Emiroglu et al. (Emiroglu et al.

2011a, b) Cd ¼ 0:836 þ �0:035 þ 0:39 Ph1

� �12:69

þ 0:158 Lb

� �0:59 þ 0:049 Lh1

� �0:42

þ 0:244Fr2:1251

� �3:018" #5:36

Table 2 Range of collected data related to the side weir discharge

coefficient

Data range Fr1 P/h1 L/b L/h1 Cd

Min 0.09 0.03 0.21 0.19 0.09

Max 0.84 2.28 3.00 10.71 1.75

Avg 0.43 0.76 1.13 3.87 0.50

STDEV 0.18 0.43 0.85 3.06 0.17

Fig. 2 A three-layer ANN architecture

Model. Earth Syst. Environ. (2016) 2:63 Page 3 of 11 63

123

often trained using back propagation algorithm. In the pre-

sent study, ANN was trained by Levenberg–Marquardt

technique because this technique is more powerful and faster

compared to the conventional gradient descent technique

(Parsaie and Haghiabi 2015a; Parsaie et al. 2015a).

Radial basis function (RBF) neural network

Radial basis function (RBF) neural network is a type of

artificial neural network widely used in image processing,

pattern recognition and nonlinear system modeling. RBFFig. 3 A RBF model structure

Fig. 4 Results of empirical formulas versus measured data

63 Page 4 of 11 Model. Earth Syst. Environ. (2016) 2:63

123

model as shown in the Fig. 3, consists of two layers, the

first layer considered as hidden layer and the second one as

output layer. The radial function is considered as transfer

function for neurons, which are in hidden layer and linear

function as output layer transfer function. Designing RBF

neural network is based on defining the center of these

functions. In other words, the aim of RBF model training is

mapping the input space to output space as f : Rn ! R.

Transfer function of the RBF model is defined as Eq. (2).

f mð Þ ¼Xni¼1

wiu m� cij jj jð Þ ð2Þ

where m is the inputs variable, wi is the weight coefficients,

u is Gaussian function, which is the basic function used as

kernel function in RBF model development and is defined

as Eq. (3).

u mð Þ ¼ e�m2

2r2

� �ð3Þ

RBF model training usually is carried out by Gradient

Descent approach. The aim of RBF model is defining the

value of kernel function parameters and weights. Initial

value of weights is defined randomly. The error for each

sample of the data set is calculated as Eq. (4).

ei ¼ ti � yi ¼ ti �XNj¼

wju mi � cij jj jð Þ ð4Þ

The error for total input data set is calculated as Eq. (5).

Table 3 Performance of empirical formulas

Author R2 RMSE

Nandesamoorthy and Thomson (1972) 0.01 0.00

Subramanya and Awasthy (1972) 0.01 0.00

Yu-Tech (1972) 0.01 0.00

Ranga Raju et al. (1979) 0.01 0.00

Hager (1987) 0.01 0.01

Cheong (1991) 0.01 0.01

Singh et al. (1994) 0.07 0.01

Jalili and Borghei (1996) 0.06 0.01

Borghei et al. 1(999) 0.11 0.02

Emiroglu et al. (2011a, b) 0.65 0.03

Fig. 4 continued

Model. Earth Syst. Environ. (2016) 2:63 Page 5 of 11 63

123

E ¼ 1

2

Xpi¼1

eij j2 ð5Þ

RBF model preparation is finished when error of RBF

model for all data sets is lower than the threshold error

which is defined by the designer (Liu 2013).

Results and discussion

Performance of empirical formulas, MLP and RBF models

was assessed by data collected the range of which is given

in Table 2. Accuracy of empirical formulas and MLP and

RBF models was assessed by statistical error indices such

Fr1

L/B

L/h1

P/h1

Cd

Input Layer

Frist hidden layer

Second hiddenlayer

Output layer

Fig. 5 Structure of MLP model

0 50 100 150 200 250 300 3500

0.5

1

1.5

Data Number

Cd

Train Data OutputsTargets

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1

1.5

R2=0.94

Measured Data

MLP

mod

el

0 50 100 150 200 250 300 350-0.2

-0.1

0

0.1

0.2

Data Number

Erorr

MSE=0.0014 RMSE= 0.03 Error

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.150

50

100

150

Error Range

Dat

a nu

mbe

r

Fig. 6 Performance of MLP model during training stage

63 Page 6 of 11 Model. Earth Syst. Environ. (2016) 2:63

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as correlation coefficient, root mean square error (RMSE),

mean square error (RMSE). It is noticeable that these

indices provide an average value for error and do not

provide any information about error distribution. All stages

of MLP and RBF models’ development have been pro-

gramed in Matlab software.

Results of experimental formulas

Performance of empirical formulas was evaluated for cal-

culating the Cdsw using data collection (Table 2) and the

results of them were compared with measured data. Fig-

ure 4 and Table 3 present the results of empirical formulas.

0 10 20 30 40 50 60 70 800

0.5

1

1.5

Data Number

Cd

Validtion DataOutputsTargets

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1

1.5

R2=0.84

Real Data

MLP

mod

el

0 10 20 30 40 50 60 70 80-0.2

0

0.2

0.4

0.6

Data Number

Eror

r

MSE=0.0045 RMSE= 0.067Error

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.40

10

20

30

Error Range

Dat

a N

umbe

r

Fig. 7 Performance of MLP model during validation stage

0 10 20 30 40 50 60 70 800

0.5

1

1.5

Data Number

Cd

Test DataOutputsTargets

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.20.40.60.8

11.21.4

R2=0.89

Real Data

MLP

mod

el

0 10 20 30 40 50 60 70 80-0.4

-0.2

0

0.2

0.4

Data Number

Eror

r

MSE=0.0032 RMSE= 0.057 Error

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.30

10

20

30

40

Error range

Dat

a nu

mbe

r

Fig. 8 Performance of MLP model during testing stage

Model. Earth Syst. Environ. (2016) 2:63 Page 7 of 11 63

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As could be seen in Table 3, most empirical formula do not

provide suitable performance. Emiroglu formula with error

indices (R2 = 64 and RMSE = 0.03) is the most accurate

one among empirical formulas.

Results of ANNs models

ANNs model development

Developing ANNs model as a common type of soft com-

puting technique is based on data set. Therefore, the col-

lected data set was divided into three groups as training,

validation and testing. Validation data set was considered

to avoid over-training of MLP model. The dimensionless

parameters presented in Eq. (1), were desirable as input

parameters for ANNs model development and discharge

coefficient was considered as model output. Data selection

for preparation of MLP model was carried out randomly.

70 % of total data set was considered for training, 15 % for

validation and the rest (15 %) for testing. Designing the

structure of MLP model is more based on the designer

experience whereas recommendation of investigators who

conducted similar researches is useful. In this paper, the

recommendation of Parsaie and Haghiabi (2015c) was

used. Preparation of ANNs model included the type of

ANNs model, number of the hidden layer(s), number of the

neurons in each hidden layer, defining suitable transfer

function for neurons of hidden layer(s), defining suit-

able transfer function for output layer and learning algo-

rithm. To obtain an optimal structure for MLP model, first,

one hidden layer was considered and then, the number of

neurons in hidden layer was increased one by one. Various

types of transfer functions such as log-sigmoid (logsig),

tan-sigmoid (tansig), linear (purelin), etc. were tested. This

process continues to obtain a model with suitable perfor-

mance. It is notable that Levenberg–Marquardt technique

was used for MLP model learning. All stages of MLP

preparation were conducted in Matlab software.

Results of MLP models

MLP model contains two hidden layers. The first hidden

layer contains ten (10) neurons with tangent sigmoid

(tansig) as transfer function and the second one contains

five neurons with Log-sigmoid transfer function. The

Fr1

L/B

L/h1

P/h1

Cd

Input Layer

Hidden layer

Output layer

Fig. 9 Architecture of RBF model

0 50 100 150 200 250 300 350 4000

0.5

1

1.5

Data number

Dis

char

ge C

oeff

icie

nt

trainoutputstrainTargets

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1

1.5R2=0.69

Measured Data

RB

F m

odel

0 50 100 150 200 250 300 350 400-0.4

-0.2

0

0.2

0.4MSE=0.0079 RMSE= 0.09

Data Number

Erro

r Val

ue

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.30

20

40

60

80

Error Range

Dat

a N

umbe

r

Fig. 10 Performance of RBF model during training stage

63 Page 8 of 11 Model. Earth Syst. Environ. (2016) 2:63

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linear transfer function was considered as neuron of out-

put layer. The structure of developed MLP model is

shown in Fig. 5. Training of MLP model was performed

with Levenberg-Marquat technique. 70 % of data set was

used for training, 15 % for validation and the rest (15 %)

was considered for testing the model. Performance of

MLP model in each stage of development (training, val-

idation and testing) is shown in Figs. 6, 7, 8 and to assess

the performance of this model, error indices for each

stage of preparation were calculated and presented in

these figures. Figures 6, 7, 8) show that accuracy of MLP

model is suitable for prediction of the Cdsw. In addition to

calculation, standard error indices the error distribution

were also plotted for the all data, which were used for

training, validation and testing. To evaluate error density,

error histogram was plotted. As could be seen in his-

togram, distribution of error is normal and more con-

centrated around zeros.

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

Data number

Dis

char

geC

oeff

icie

nt

valoutputsvalTargets

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1

1.5R2=0.70

Measured Data

RB

F m

odel

0 10 20 30 40 50 60 70 80 90 100-0.4

-0.2

0

0.2

0.4MSE=0.00635 RMSE= 0.0797

Data Number

Erro

rVal

ue

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.30

10

20

30

Error Range

Dat

a N

umbe

r

Fig. 11 Performance of RBF model during validation stage

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

Data number

Dis

char

ge C

oeff

icie

nt

testoutputstestTargets

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1

1.5

R2=0.71

Measured Data

RB

F m

odel

0 10 20 30 40 50 60 70 80 90 100-0.4

-0.2

0

0.2

0.4

MSE=0.00635 RMSE= 0.0797

Data Number

Erro

r Val

ue

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.30

10

20

30

Error Range

Dat

a N

umbe

r

Fig. 12 Performance of MLP model during validation stage

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Results of RBF models

To assess the accuracy of RBF model to predict Cdswand compering its performance with MLP model, it has

been attempted to hold similar conditions for model

development. In other words, it has been attempted to

hold similar number of the training, validation and

testing data set. In addition, it has been attempted to

hold similar number of neurons in input layer. The

architect of RBF model is shown in Fig. 9. As shown in

Fig. 9, the input layer neurons were considered equal to

MLP model input later. Performance of RBF model

during training, validation and testing model is shown in

Figs. 10, 11, 12. As shown in these figures, performance

of RBF model to predict Cdsw is not suitable. Comparing

performance of RBF model with MLP model in training,

validation and testing stages shows that MLP model is

more accurate.

Conclusion

In this study, side weir discharge coefficient (Cdsw) was

calculated and predicted by empirical formulas and radial

basis function (RBF) neural network along with multilayer

perceptron (MLP) neural network. Results of this study

indicate that Emiroglu formula is the most accurate one

among empirical formulas. To achieve more accuracy in

Cdsw prediction, MLP model and RBF model were devel-

oped and to prepare MLP and RBF models, about 477 data

sets related to Cdsw were collected. Results of assessing

performance of MLP show that MLP model has suit-

able performance to predict Cdsw. Results of RBF model

development indicated the accuracy of this model is a little

better compared to empirical formals. In general, perfor-

mance of MLP model is much more compared to RBF and

empirical formulas.

Compliance with ethical standards

Conflict of interest The authors declare that there is no conflict of

interest regarding the publication of this manuscript.

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